Analysis of exclusively kinetic two-link underactuated mechanical systems

Abstract

Analysis of exclusively-kinetic two-link underactuated mechanical systems is undertaken in this paper. It is first shown that such systems are not full-state feedback linearizable around any equilibirium point. Also, the equilibrium points for which the system is small-time locally controllable (STLC) is at most a one dimensional submanifold. A concept less restrictive than STLC, termed the small-time local output controllability (STLOC) is introduced, the satisfaction of which guarantees that a chosen configuration output can be controlled at its desired value. It is shown that the class of systems considered are STLOC, if the inertial coupling between the input and output is non-zero. Also, in such a case, the system is nonminimum phase (NMP). An example section illustrates all results presented.

Dynamics of Underactuated Multibody Systems

“…As shown in Isidori [44] from these properties follows the identity (3.9) where f , ω are vector fields and λ is a scalar function.…”

“…Independent means in this context that the differentials of λ j are linear independent. As shown in [44] the existence of such a solution is equivalent to the complete integrability of the distribution Δ.…”

“…In the case of a bounded internal dynamics, linear control techniques can be used for the linear subsystem. The various aspects, developments and results of nonlinear control systems based on feedback linearization are summarized in extensive detail in the reference books of Isidori [44] and Nijmeijer and van der Schaft [57]. More compact introductions to feedback linearization are found in the textbooks Sastry [64] and Slotine and Li [66] and in the survey paper of Krener [53].…”

“…The following introductional representation of feedback linearization and model inversion is mainly based on the reference books Isidori [44], Sastry [64] and Slotine and Li [66]. The goal is to present this topic in sufficient depth to understand the application of these concepts to underactuated multibody systems in the later chapters.…”

“…An introduction to differential geometry can be found in compact form in the reference books concerning nonlinear control by Isidori [44], Marino and Tomei [55] and Nijmeijer and van der Schaft [57]. An extensive treatment of differential geometry and smooth manifolds is given in books solely dedicated to those topics such as, e.g., Burns and Gidea [6], Lee [54], Olver [58] and O'Neill [59].…”